A Metanilpotency Criterion for a Finite Solvable Group

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ≅ A 5.

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we generalize a result of J. G. Thompson and characterize finite solvable groups as those finite groups which do not contain nontrivial elements xi , i = 1,2,3, with x 1 x 2 x 3 = 1 and xi a pi -element for distinct primes pi . We note some connections with coverings of curves.

Let A be a finite group and let n be the number of primes, including multiplicities, which divide the order of A. Suppose further that G is a finite solvable group which admits A as a fixed-point-free operator group and such that (i A I, G|I ) = 1. Under these assumptions it has been conjectured that l(G), the nilpotent length of G, is at most n, while lp(G), the p-length of G, is at most [(n+1)/2]. These inequalities have been verified and shown to be best-possible in certain special cases ( [3 ][5], [1]). It is the purpose of this note to show that these conjectured upper bounds are actually obtained provided A is solvable. Specifically we will prove

Finite Groups With Many Conjugate Elements

Blocks of small defect

Regular and p-regular orbits of solvable linear groups

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

The main result in this note is the followingTheorem: Let G be a finite solvable group. There exists a permutation σ of the set G such that {g • σ(g); g∈G} = G if and only if the Sylow 2-subgroup of G is non-cyclic or trivial

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \({\mathfrak {m}_{p}(G)}\) , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.

It is proved that finite groups with 10p elements of the largest order, wherep ≥ 23 a prime, are solvable in this article. Consequently it follows that all finite group with 10p elements of largest order are solvable by some other results.

Let F be a free group of arbitrary rank and let H be a finitely generated subgroup of F. Given a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Our main result states that it is decidable whether H is pro-Su dense, where Su⊂S denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether H is pro-S dense?

Minimal sumsets in finite solvable groups

A subgroup H of a finite group G is said to be “semi-cover-avoiding in G” if there is a chief series of G such that H covers or avoids every chief factor of the chief series. In this article, some new characterizations for finite solvable groups are obtained based on the assumption that some subgroups have semi-cover-avoiding properties in the groups.